aa r X i v : . [ h e p - ph ] J u l Applied Holography of the AdS -Kerr Spacetime Brett McInnes
National University of Singaporeemail: [email protected]
Asymptotically Anti-de Sitter Kerr black holes (we focus here on the five-dimensionalcase) are associated holographically with matter at conformal infinity which has a non-zeroangular momentum density. It is natural to attempt to associate this angular momentumwith the recently discovered vorticity of the plasmas produced in peripheral heavy-ioncollisions. We assume that an AdS -Kerr black hole with angular momentum to massratio A is dual to boundary matter with an angular momentum density to energy densityratio also equal to A . With this assumption, we find that, for collisions corresponding toa given value of A , there is a maximal possible angular velocity (well below the maximalvalue permitted by causality) for such matter at infinity, and that this value is in approx-imate agreement with the experimentally reported value of the average plasma vorticityproduced in typical peripheral collisions of heavy ions. . Black Hole Angular Momentum and its Holographic Dual There is a clear sense in which a generic electrically neutral astrophysical black hole isrepresented by the asymptotically flat Kerr metric in four dimensions: the Schwarzschildmetric only occurs as an extremely special case. In the five-dimensional asymptoticallyAdS case, the role of the generic black hole metric (with a topologically spherical eventhorizon) is played by the AdS -Kerr metric [1]; the corresponding spacetime merits closeattention on those grounds alone, and in fact this statement holds true in a more specificsense, as follows.In its most familiar and best-studied form, the holographic or gauge-gravity duality posits that physics in an asymptotically AdS spacetime is dual to that of an N = 4 super-Yang-Mills theory, with a large number of colours, defined on the four-dimensionalconformal boundary. There is some reason to hope that this kind of field theory can shedsome light on the behaviour of the Quark-Gluon Plasmas (henceforth, QGP) producedin collisions of heavy ions [2]. Since such plasmas equilibrate very quickly and so havewell-defined temperatures, one focuses on bulk systems with similarly well-defined tem-peratures, that is, on asymptotically AdS black holes with large Hawking temperatures.Most collisions of this kind will be measurably peripheral, that is, off-centre to somenon-trivial degree (measured by the parameter known as centrality .) In such a colli-sion, a large quantity of angular momentum is transferred to the QGP [3–5], and so thegauge-gravity dual must likewise have a large angular momentum. Clearly, then, we needto understand the holography of the AdS -Kerr black holes : these represent the generic case in this application, just as the asymptotically flat Kerr black hole is generic in theastrophysical application .This need has been underlined by the very remarkable recent observations made bythe STAR collaboration at the RHIC facility, who have reported indirect but convincingevidence [6–9] of local rotational motion (“ vorticity ”) in the QGP produced in peripheralcollisions of gold nuclei at various impact energies. (The vorticities are deduced fromobserved polarizations of Λ / Λ hyperons: see [6] for a clear discussion of the experimentand of the discovery.)Now these “vortical plasmas” are extremely complex systems, and one might well bepessimistic as to the prospects for establishing a comprehensive duality between them theAdS -Kerr black hole, characterized as it is by a very small number of parameters. Infact, even aside from any holographic interpretation, rotating quantum-chromodynamicsystems are not simple to describe or even to define, and this in itself is a matter of currentresearch: see for example [14], where a lattice approach is found useful, and [15], wherethe powerful analogy between rotation and the effects of magnetic fields is exploited.On the holographic side, the most basic objection to such an enterprise is based onthe fact that the spatial sections of the conformal boundary are not flat, as they are inmost other applications of gauge-gravity duality (see however Chapter 14 of [2]). This The gauge-gravity duality for rotating (topologically spherical) AdS black holes has been studied,and successfully applied, previously: see [10, 11]. These works explain in detail how the AdS/CFTcorrespondence works in the rotating case. They deal only with a four-dimensional bulk geometry, butthere should be no difficulty in adapting to the five-dimensional case. (In the case of a four-dimensionalbulk, one must use the duality of a system of N c M2-branes with a boundary theory defined on a three-dimensional spacetime. See [12, 13].) -Kerr metricitself is already a formidably complex object: allowing the geometry to be dynamic willcertainly be necessary if a truly realistic model of the vortical plasma is to be constructed,but this is a project for the future.Here, as an initial step, we set ourselves a much more modest task: can we show thatthe boundary matter dual to the AdS -Kerr black hole behaves in a manner that is evenmoderately realistic when compared with the actual vortical QGP? In particular, can weconstruct a holographic model which predicts, at least up to order of magnitude, the mainparameter reported in [6], the average vorticity ω , given there as ω ≈ ± × · s − ?We will argue that this can indeed be done: within the (admittedly large) uncertainties,the AdS -Kerr model does predict values for the (suitably interpreted) average (over thevolume of the plasma) vorticity, at each impact energy, which agree with those reportedby the STAR collaboration. (The agreement is good at high impact energies, less goodat lower impact energies.) The predicted average over impact energies is . × · s − ,which, again, in view of the large uncertainties on both the observational and theoreticalsides, is acceptable.Let us proceed to review the AdS spacetime and its conformal boundary.
2. The AdS -Kerr Geometry: Bulk and Boundary The AdS -Kerr metrics were given in [1] (but see also [21] and [22] for important discus-sions of the formulae for the physical mass and angular momentum), in the case wherethe black hole is electrically and magnetically uncharged . In five dimensions, the blackhole can rotate around two distinct axes simultaneously, so in general one has a pair ofrotation parameters, ( a, b ) ; but here, for simplicity, we set the second rotation parameter, b , equal to zero. The AdS -Kerr metric in this simplified case takes the form g (cid:16) AdSK ( a, (cid:17) = − ∆ r ρ " d t − a Ξ sin θ d φ + ρ ∆ r d r + ρ ∆ θ d θ (1) + sin θ ∆ θ ρ " a d t − r + a Ξ d φ + r cos θ d ψ , In principle we can also endow the black hole with electric charge, as in [23], in order to modela non-zero baryonic chemical potential. However, just as the metric at conformal infinity for the four-dimensional AdS-Kerr-Newman metric is formally independent of the electric charge, so also the inclusionof electric charge in this case will not modify the form of the boundary metric given in equation (7) below(which is all we need in this work). Note in this connection that the rate of fall-off of the electriccontribution to the metric is larger in five than in four dimensions. ρ = r + a cos θ, ∆ r = ( r + a ) (cid:16) r L (cid:17) − M, ∆ θ = 1 − a L cos θ, Ξ = 1 − a L . (2)Here L is the asymptotic AdS curvature length scale, t and r are as usual, and the angularcoordinates θ, φ, ψ on the topological three-sphere will be described in detail below. Theparameters a and M should be regarded strictly as quantities describing the geometryof the spacetime; they are related ( but by no means equal ) respectively to the blackhole specific angular momentum (angular momentum per unit physical mass) A and thephysical mass m .In fact, setting b = 0 in the formulae given in [22], we have (if j denotes the blackhole’s physical angular momentum) m = πM (2 + Ξ)4 ℓ B Ξ , j = πM a ℓ B Ξ , (3)where ℓ B is the gravitational length scale in the bulk (which is unrelated to the Plancklength in physical spacetime).One should note here that there are several possible distinct definitions of the massof the black hole; this is discussed in detail in [22]. In particular, in five dimensions onehas to decide whether to include the contribution of the Casimir energy at infinity. Thisis appropriate in some applications (certainly if one is interested in the holography of theconformal anomaly [24, 25]), but not in the application to the physical QGP, which weare attempting to describe here. Therefore we follow [22] (see in particular their Footnote5), where the constant of integration arising in the First Law is systematically set equalto zero, so that empty AdS has zero mass, and there is no Casimir contribution to theenergy density of the boundary field theory.From the equations (3) we see that the angular momentum to (physical) mass ratio,or specific angular momentum, is given by A = 2 a a − ( a /L ) . (4)Note carefully that this differs from the four-dimensional case, where a itself is the specificangular momentum.Other important characteristics of the black hole can be computed from its geometry:for example, the Hawking temperature is given [22] by T = r H (cid:16) r H L (cid:17) π ( r H + a ) + r H πL , (5) In the system of natural units we use here, a has units of length, while M has units of squared length. In natural units, A has units of length, m of inverse length. r H denotes the horizon radius (which can be regarded as a function of M and a through its definition as the largest root of ∆ r ), and the entropy by S = π ( r H + a ) r H ℓ B Ξ . (6)Finally, we note that the angular coordinates θ, φ, ψ used in [1], which we followhere, are not the familiar polar coordinates on the three-sphere: for example, one seesthat, when A = 0 , the angular part of the metric is r (cid:0) d θ + sin θ d φ + cos θ d ψ (cid:1) ,which is indeed the usual round metric on S with radius r , but not in polar coordinates.Instead, these are the coordinates usually used to describe S when it is regarded asa principal U (1) -bundle over S , that is, as the Hopf bundle [26]. The two coordinates φ and ψ both run from to π , while θ runs from to π/ ; thus in fact each fixedvalue of θ corresponds to a two-torus, except in the “degenerate” cases θ = 0 , π/ , whichcorrespond to one-dimensional circles. In particular, the single condition θ = π/ reducesthe dimensionality by two : it means that we are on the equator of S , described by asingle angular coordinate, φ .The geometry of the conformal boundary is fixed by means of a conformal re-scalingof the metric g (AdSK ( a, ) as the limit is taken to infinity. We take the boundary metricto be g (cid:16) AdSK ( a, (cid:17) ∞ = − d t + 2 a sin θ d t d φ Ξ + L d θ − ( a/L ) cos θ + L sin θ d φ Ξ + L cos θ d ψ , (7)obtained from g (AdSK ( a, ) by extracting a conformal factor r /L , taking the limit r →∞ , and then doing some algebraic simplifications. We have chosen the conformal factorso that the time coordinate t represents proper time for a stationary observer at infinitylocated at one of the poles ( θ = ψ = 0 ); we can take this observer to be an outside observer,fixed in the laboratory. In each case we consider below (particles with zero, respectivelynon-zero angular momentum), d φ/ d t represents an angular velocity as measured by thisobserver . In terms of the usual coordinates ( x , x , x , x ) on Euclidean IR , we have, for a three-sphere ofradius r , x = r cos ψ cos θx = r sin ψ cos θx = r cos φ sin θx = r sin φ sin θ. Notice however that, if we set ψ = 0 , then the remaining coordinates can be interpreted as polar coordi-nates on a hemisphere of S . In view of the “large” angular velocities arising in our application, it is natural to ask whether weshould give here a relativistic account of angular velocity. The answer is that, while the angular velocitieshere seem large by ordinary standards, they are in fact surprisingly small in view of the size of the systemsin question. Perhaps the best way to see this is to use natural units, taking the femtometre as the basicunit. In these units, we will find later that a typical angular velocity here is on the order of .
004 fm − .For systems with a radius of a few femtometres, this does not lead to relativistic velocities, so we willnot take relativistic effects into account here. See [27] for a discussion of this. Another way of seeingthe point is to note that, during the lifetime of the plasma, a system rotating at such angular velocitiesexecutes far less than one complete revolution [28].
5e stress, because it will be crucial later, that the physical parameters of the AdS -Kerr black hole (the physical mass m , and the specific angular momentum A ) are relatedto the geometric parameters M and a in a surprisingly indirect manner (equations (3)and (4) above). This is due to the ubiquitous presence of the quantity Ξ , which appearsboth in the bulk metric and in its counterpart on the conformal boundary. This in turnis required in order to maintain the regularity of the geometry in both cases .Before proceeding, we draw the reader’s attention to the following (ultimately) veryimportant aspect of the AdS -Kerr geometry. Examining the metric, we see that, if noconstraint is imposed on a , then the signature of the coefficient of d θ , / ∆ θ , can vary,depending on θ : it is positive for θ > arccos ( L/a ) , but negative for θ < arccos ( L/a ) .At θ = arccos ( L/a ) there is a “singularity”, which admittedly proves to be a coordinate“singularity”; nevertheless, it seems clear that this kind of behaviour is a complicationwhich we should consider only if it cannot be avoided. We therefore require, in this work,that a should always be smaller than L .In fact, it is natural to impose a stronger condition on a , for the following reason. Thestatus of cosmic censorship in higher-dimensional spacetimes is currently unsettled: see forexample [29, 30] and references therein. In discussing gauge-gravity duality, however, weclearly need to assume that some form of censorship does hold, since the boundary theory(representing an equilibrated plasma) certainly does have a well-defined temperature andentropy density; and so the dual object in the bulk must have a well-defined event horizon.For the geometry we are discussing here, censorship takes the form M ≥ a , or, expressedin terms of the physical mass, ℓ B m Ξ ≥ π a (2 + Ξ) ; (8)we see from this that, if m takes any fixed finite value, then (because of the factor of Ξ on the left) this condition will be violated, for some a strictly smaller than L , if westeadily increase a from zero towards L . With these assumptions, then, a cannot evencome arbitrarily close to L , much less attain that value. (See [31] for a discussion of thisin the four-dimensional case; see also [32].) This is a useful piece of information, for itmeans that, in all of the many expressions depending on the reciprocal of Ξ (or its squareroot), we are not dealing with quantities which can be arbitrarily large.The physics of the AdS -Kerr black hole determines, according to the gauge-gravityduality [2], that of a field theory on the boundary ( r → ∞ ); this field theory is heldto approximate, to some extent, to the QGP. Thus, the Hawking temperature T of theblack hole corresponds to the temperature of the plasma, the ratio of the black hole’sentropy to its (physical) mass is equal to the ratio of the plasma entropy density s to itsenergy density ε , and similarly the black hole specific angular momentum parameter A isinterpreted as the ratio of the QGP angular momentum density α to its energy density.Finally, the bulk curvature length scale is in principle given a dual interpretation on theboundary by the “holographic dictionary”; it is related to the number of colours and the This is particularly clear in the boundary geometry: there, the circumference of a circle of the form θ = θ = constant, located on the two-dimensional hemisphere ψ = 0 , is πL sin θ / √ Ξ , while its radius,measured from the pole (of both the 3-sphere and the two-dimensional hemisphere) along the hemisphere,is L R θ dθ √ − ( a /L ) cos θ ; the ratio only tends to π as θ → because the √ Ξ factor is present.
6t Hooft coupling of the boundary field theory . In practice, L is not known, but its valuecan be usefully constrained, for example as follows.Notice first that L cannot be scaled away here, since the event horizon is topologicallyspherical. This is not a new observation (see [2], Chapter 14, for a detailed discussionin the non-rotating case), but it is important, because it is related to the fact that theboundary field theory is defined on a compact space, whereas of course the actual spacewhich the plasma inhabits is not compact. In principle, this leads to various propertieswhich may not be welcome or realistic (for example, perturbation spectra of fields onthe black hole background become discrete, there is a phase transition (the Hawking-Page transition) which may not however correspond well with the actual phase transition(hadronization) experienced by the plasma, and so on). It follows that the use of thetopologically spherical Kerr geometry in holographic models (as in [10, 11] and here) isreally only acceptable if the volume of the compact space is very large relative to the sizeof the system being studied. As we now explain in detail, in our case this is in fact impliedby our discussion above.The spatial manifolds defined by t = τ = constant on the boundary have the geometryof a deformed three-sphere, with metric g (cid:16) AdSK ( a, (cid:17) ∞ ( t = τ ) = L d θ − ( a/L ) cos θ + L sin θ d φ Ξ + L cos θ d ψ . (10)The volume of the three-sphere with this metric is given by V ( L, A ) = 2 π L (cid:20) L a (cid:18) √ Ξ − (cid:19)(cid:21) , (11)where, by inverting equation (4), we regard a as a function of A and L (and hence we cando the same for Ξ , the last member of (2)). We have written the volume in this form soas to enable a comparison with the volume of the sphere in the case where A → , whichis of course π L .Now, our condition that a should be strictly smaller than L implies that Ξ must bepositive, so it is clear from equation (4) that A is always smaller than a , and thereforethan L . In short, we always have A < L. (12)Since a is smaller than L , the expression in square brackets in equation (11) is larger thanunity for all non-zero A ; so if we use π L to compute the volume, we will under-estimateit (by, as it will turn out, a factor of about 3). On the other hand, for collisions of gold ionsat 200 GeV impact energy per pair and 20 % centrality, we find below that a reasonableestimate of the ratio of the plasma angular momentum density to its energy density is ≈
72 femtometres, and in the holographic model this is A ; so L should be larger than this. In detail, one has ℓ B L = π N c , ℓ s L = 1 λ , (9)where N c is the number of colours in the boundary field theory, λ is the ’t Hooft coupling in that theory,and ℓ s is the string length scale. The duality is useful only when the bulk can be treated classically andwhen strings can be treated as point particles: that is, when L is large relative to the other two lengthscales. These conditions are certainly satisfied here, with the lower bound on L we are about to describe.
7n this way we obtain an extremely conservative lower bound for the spatial volume inthis case: V ( L, A ) ≥≈ . × fm ; the real figure is probably an order of magnitudelarger. At lower impact energies, the volume computed in this way is somewhat smaller,but, in all cases where vorticity has actually been detected experimentally, it is essentiallyinfinite compared to the size of the system we are treating. (The region occupied by theequilibrated plasma resulting from a collision at this centrality has a total volume on theorder of 100 fm .) We do not therefore anticipate any difficulties on this score.We now turn to our main problem: can we compute the angular velocity of the matterat infinity which is dual to an AdS -Kerr black hole with a specified ratio A of angularmomentum to mass? That is, can we give a holographic estimate of the vorticity of a“QGP-like” fluid in terms of the ratio α/ε of the plasma’s angular momentum density toits energy density?
3. The AdS -Kerr Geometry: Angular Velocity at Infinity Our task now is to use the AdS geometry to estimate angular velocities on the boundary.This is not quite straightforward.The black hole introduced above, with an angular momentum per unit mass valueequal to A , is dual to a fluid on the boundary, described by a field theory in the usualholographic manner. Since the duality is a complete equivalence, we assume that thissingle parameter, A , sets the scale for all rotational phenomena in the boundary theory,as it clearly does in the bulk. This paucity of parameters is an indirect consequence of the“no-hair” theorems, which indeed dictate that the bulk geometry is determined by a verysmall number of physical quantities, including (in our case) a single angular momentumparameter .This means that we are committed to a matter model on the boundary in which thereis only one angular momentum scale, which of course the holographic correspondencedictates should be equal to A . Similarly there is only one angular velocity scale; theboundary matter rotates like a rigid body.The detailed way in which vorticity develops in the actual plasma is very complex.The vorticity is thought to be initially focused mainly in a thin layer, with a thicknesspresumably measured in fractions of a femtometre, between the participant and spectatormatter [34, 35]; it then propagates inward, through viscous effects.The motion of this system cannot, of course, be literally pictured as a simple rotatingobject. When [6] characterizes this system by an angular velocity given as ± × · s − (with a certain systematic uncertainty), the intention is not, of course, to claim that thematter in any actual QGP vortex rotates at this rate. Instead, this quantity is intendedto give an overall, averaged (over the volume of the plasma, over the full range of impactenergies, and so on) indication of the internal motion of the vortical plasma. One shouldin fact interpret this number as nothing more than a measure of the extent to whichthe internal dynamics of the vortical plasma differs from that of the plasmas produced incentral collisions . Our task here is to try to use a computation of the angular velocityat infinity for the AdS -Kerr spacetime to reproduce this number (and the allied numbers In the higher-dimensional context, “no-hair” statements apply when one fixes the topology of theevent horizon to be spherical, as we are doing here. See [33]. A ,having equatorial orbits. The boundary metric, given in equation (7), has a Killingvector field proportional to ∂ φ (the normalization being fixed by our assumption that φ has periodicity π ). The inner product of this vector field with the unit tangent to theparticle worldline, ˙ t ∂ t + ˙ φ ∂ φ (where dots denote differentiation with respect to the propertime of the particle), gives us the angular momentum per unit mass.Let us begin by considering such particles with zero angular momentum. Computingthe inner product as above, we have in that case ˙ ta Ξ + ˙ φL Ξ = 0 , (13)from which we have at once, denoting the angular velocity of these particles by ω , ω = − aL . (14)Because of this equation, it is sometimes said that “the boundary rotates with angularvelocity − a/L ”. But this is just a way of describing the motion of particles with adistinguished angular momentum to mass ratio, namely zero. Here however we are notinterested in such particles: we wish the particles to have an angular momentum to massratio A , corresponding holographically to the bulk black hole with angular momentum tomass ratio having that same value.We propose that the physical angular velocity ω , corresponding to the vorticity of theplasma as observed experimentally, should be computed as the difference between theangular velocity of particles with non-zero angular momentum, ω A , and that of fictitiousparticles with zero angular momentum, ω : we have ω = ω A − ω . (To put it morepicturesquely: we compute the angular velocity by using a frame at infinity which “co-rotates with the boundary”.)To compute ω A , we have again, from equation (7), ˙ ta Ξ + ˙ φL Ξ = A . (15)We need to supplement this with the fact that ˙ t∂ t + ˙ φ ∂ φ is a unit vector in the boundarygeometry, that is, − ˙ t + 2 ˙ t ˙ φ a Ξ + ˙ φ L Ξ = − . (16)Using this equation, we can eliminate ˙ t , so obtaining a quadratic equation for ω A : L ω A + 2 a ω A + a L − A a Ξ A L Ξ ! = 0 . (17)Solving this and subtracting ω as suggested, we obtain two equal and oppositely signedvalues for ω , corresponding of course to the two possible directions of rotation; taking the9ositive value we have finally, ω = A L s Ξ1 + A L Ξ , (18)where, through equation (4), we can regard Ξ as the function of A given by Ξ = q A L − − A L A L . (19)These are the relations we seek: ω is interpreted as the vorticity of the plasma, and,if that plasma has angular momentum density α and energy density ε , we have α/ε = A .The vorticity of the plasma is now computed in terms of its physical parameters (togetherwith L , see below.)The relation between ω and A appears to be unusual: note for example that ω is smallwhen A is sufficiently large. Of course, unusual relations between angular velocity andangular momentum are not unexpected for matter associated with rotating black holes:“frame dragging” has the same origin. In this specific case, one way to understand thestructure of equation (18) is to observe that it ensures that causality is never violated here.To see this, let us compute the linear velocity of objects on the equator of the boundarysphere, having angular velocity ω . From equation (7) we see that the circumference is πL/ √ Ξ , which is larger than πL , by a factor which diverges as a approaches L — soindeed large values of A (and a ) might easily give rise to very high linear velocities in thiscase. For an object with angular velocity ω , the linear velocity of this potentially largecircumference is given by v ω = ωL √ Ξ , (20)and this could exceed unity, violating causality, as a and A approach L , unless the depen-dence of ω on a causes it to tend to zero sufficiently rapidly in the same limit . In short,the angular velocity must decrease towards zero when the parameter a is sufficiently large,in order to avoid violating causality.In fact, substituting equation (18) into (20) we find v ω = A L s
11 + A L Ξ , (21)and (in view of inequality (12) and the fact that Ξ is always positive here) this does alwayssatisfy causality, because the factor of √ Ξ in the denominator of the right side of (20) hasbeen cancelled.In principle, equations (18) and (19) allow us to make a holographic prediction oftypical angular velocities characterizing a vortical plasma in terms of the ratio of its Of course, we know that (as long as the physical mass of the black hole remains finite) a cannotactually come arbitrarily close to L without violating cosmic censorship. We cannot rely on this to avoidcausality violation here, however, for the following reason. Censorship is expressed for this black holeby the inequality (8), which involves, on the left, the quantity ℓ B . Unfortunately we do not know thisquantity (apart from the fact that it must be small relative to L , see the holographic “dictionary”, above),so in practice we cannot say precisely how large a can be relative to L . α to its energy density ε . In practice, unfortunately, this isnot possible: for it is clear that the relation is mediated by L , which we do not know. As inthe computation of the spatial volume at infinity, we have to find a way of circumventingthis.
4. The Bound on ω , and a Conjecture The black hole parameter A is computed holographically from the ratio α/ε , which can beestimated explicitly from physical data and phenomenological models, and we will explainhow to do this shortly. Fixing A at a definite value in this manner, we may use equations(18) and (19) to regard ω formally as a function of L , ω ( L ) . We now ask: what form doesthis function take?We stress here that, in doing so, we are not “varying” L ; we are merely exploringwhether it is possible to deduce anything regarding ω without knowing what value L actually takes.We know (from the inequality (12)) the domain on which L is defined: it is the openinterval ( A , ∞ ) . Clearly ω vanishes as L → ∞ , as expected, since ω ≈ A /L in thatlimit: this is like a system with a large moment of inertia. We have seen that it also vanishes as L tends down to A , for reasons connected with causality. Now since ω iscontinuous and positive as a function of L , it follows that ω is bounded above . This meansthat it is possible at least to put an ( A -dependent) bound on ω without knowing L .In fact, ω ( L ) can be completely described analytically; the analysis is rather intricatebut of course essentially elementary. One finds that this function has a unique maximum, ω max , given by ω max = κ A ≈ . A , (22)where κ is a pure number which can be computed to any desired precision, having theindicated approximate value. Clearly, no matter what value L actually takes, ω can neverexceed this value, once A has been fixed.Notice that, even when the particles in the model with angular velocity ω have themaximal possible angular velocity for a given value of A , they move at a velocity v ω ( ω max ) , computed using equation (21), from which we obtain v ω ( ω max ) ≈ . , (23)that is, well below light speed. Thus, while we saw earlier that our model is alwaysconsistent with causality, the latter does not explain the existence of this maximal angularvelocity for given A — an important point . One regards ω A as a function of the dimensionless variable σ = L/ A , computes the derivative andsets it equal to zero; a somewhat elaborate algebraic manipulation reduces this condition to solving theoctic σ + 63 σ − σ − σ + 48 = 0 . This is a quartic in σ and so it can be solved explicitly andexactly if desired. One finds that there are two real positive solutions for σ , of which however only one,given approximately by σ ≈ . , allows the crucial inequality (12) to be satisfied. Substituting thisvalue of σ back into equations (18) and (19), one obtains the maximum possible value of ω , expressed asa multiple of / A . Of course, in general, the question of causality can indeed be crucial in discussions of (sufficientlylarge) rapidly rotating systems: see for example [36] and references therein. The point is that this is notthe case here.
11n terms of the physical quantities, we have the bound ω ≤ κ εα ≈ . εα . (24)This is of interest for three reasons. First, of course, we can evaluate the right side usingphenomenological models, and compare it with the reported data on vorticity; this gives usa new way of testing holography in general. Again we stress that peripheral collisions are generic : if holography fails this test, then its general validity will be called into question.Secondly, this inequality identifies for us the “right” variable to be examined in studiesof the vortical plasma, namely ε/α . Just as realistic, astrophysical black holes are conven-tionally described by their angular momentum to mass ratio, so here holography identifies(the reciprocal of) that ratio as the one on which we should focus attention. Notice inthis connection that α and ε , as densities of conserved quantities, can be expected to varystrongly as the plasma expands; but their ratio is expected to be much more stable.Finally, and related to this second point: the inequality focuses our attention on aspecific value of ω for each peripheral collision, namely . ε/α . This specific value isnow strongly distinguished , and we should expect it to play some special role. In fact, wecan speculate that it appears here as the right side of an inequality merely as an artefactof our ignorance of L . Perhaps we should expect that . ε/α is close to the actual value of the vorticity, for impact parameters which are not very small or large (see below).In short, we conjecture that the vorticity is actually given holographically as ω = κ εα ≈ . εα (25)for a range of impact parameters or centralities to be described below.This conjecture can now be tested.
5. The Data
Before we begin, we should be open regarding the fact that precision is not to be looked forin these computations. As is well known, the field theories considered in the gauge-gravityduality are not (in all regimes) closely similar to QCD: there are important analogies(discussed very clearly in [2]) but there are basic differences. Furthermore, as we discussed,the no-hair theorems strongly restrict the number of parameters available to describethe dual plasma: we have only one angular momentum parameter, but Λ / Λ hyperonpolarization has a longitudinal component apart from the component we study here; themodel cannot describe this. The observational data, too, suffer from large uncertainties: [6]reports a large systematic uncertainty in the given values of the polarization percentages.(The principal difficulty here is that, to relate vorticities, which are not directly observable,to polarizations, one needs to know the precise temperature corresponding to a givenenergy density; and this is a notoriously difficult problem [37].)Rather than continuously repeat these points, we will simply proceed to a detailedcomparison with the data; our main priority, however, is always to assure ourselves that(24) and (25) impose constraints on or predict values for hyperon polarizations that areat least of a reasonable order of magnitude. Our secondary objective is to explain clearly12ow the relevant calculations can be done when more precise data, and perhaps moresophisticated holographic models, become available in future.We need estimates for the angular momentum of the QGP in peripheral collisions, andalso for the energy density, for various impact energies and centralities. For the former,we rely on [38], where estimates using the AMPT (“A Multi-Phase Transport”) modelare given; for the latter we use [39], where very detailed computations of many relevantparameters (using a colour string percolation model) are given. In both cases, the valuesgiven seem reasonable. Other models might be chosen, but the differences are unlikely tobe sufficiently large as to modify our general conclusions.Proceeding in this way, we can (with simple additional assumptions regarding thegeometry of the overlapping nuclei) compute the right side of the inequality (24), andthus place an explicit bound on the vorticity in each case. We can relate this to theaverage polarizations of primary Λ and Λ hyperons by using the same equation as in [6], P Λ ′ + P Λ ′ = ωT , (26)where natural units are used and T is the temperature as usual. (Actually, we will useit “in reverse”: in [6], it is used to deduce vorticities from polarizations, here we do theopposite.) This equation allows us to express the vorticity bound as a bound on the totalpolarization, obtaining inequalities of the form (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN , C ) ≤ Φ( √ s NN , C ) ≡ κ εαT , (27)where C denotes centrality and where we now know how to compute Φ( √ s NN , C ) fromphenomenological estimates in each instance; it is expressed as a percentage.We consider three different impact energy regimes. Let us begin by considering the highest-energy collisions studied by the STAR collab-oration, particularly those with an impact energy of 200 GeV per pair, for which theobservation of vorticity [7] is most unambiguous.In [6, 7], the focus is on collisions with to centrality. This means [41, 42] thatthe impact parameters vary from around 6.75 femtometres (fm) up to around 10.5 fm.On this domain, one finds [38] that the angular momentum imparted to the plasma in 200GeV collisions steadily decreases, from about (in natural units; in conventionalunits, · ~ ) at b = 6 . fm to around at b = 10 . fm. However, the volume ofthe overlap region also decreases as the impact parameter increases, and we find that, to agood approximation, the two effects cancel for collisions with to centrality: thatis, the angular momentum density α is roughly independent of b in this range of b values.We therefore focus on collisions at centrality, since, in this range, these collisions areleast affected by the variations of the nuclear density near the boundary of the nucleus, Note that, in [38], the convention is used in which ω is twice as large as in [6] and here. The primes indicate that these are the polarizations for “primary” hyperons, which means that weare neglecting the “feed-down” effect; see [40] for the theory of this, and [6] for a discussion of its effecton the STAR observations. √ s NN = 200 GeV collisions at C = 20% centrality, the angular momentum density is approximately given by α ( √ s NN = 200 GeV , C = 20%) ≈
758 fm − . (28)According to [39], the energy density in this case is approximately . / fm , and sowe compute the maximal vorticity, according to equation (22), as ω max ( √ s NN = 200 GeV , C = 20%) ≈ . − . (29)In order to compute a total polarization from this, we need to use the (initial) temperatureof the plasma, and this is the point of greatest uncertainty, as mentioned above. Usinga temperature of approximately 190 MeV [39] (which may be an over-estimate, so ourresult may well be somewhat too low), we can express the vorticity bound in this case inthe form (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 200 GeV , C = 20%) ≤ ≈ . . (30)When the first observations of polarization of Λ and Λ hyperons were announced, such avalue was too small to be detected (see the rightmost points in Figure 4 of [6]). Subsequentanalysis of a much larger data set [7] has however found evidence of such polarization,reporting values of P Λ ′ ( √ s NN = 200 GeV , C = 20%) ≈ . ± .
040 (+ 0 . − . and P Λ ′ ( √ s NN = 200 GeV , C = 20%) ≈ . ± .
045 (+ 0 . − . , the uncertainties being statistical and systematic respectively.This is the most precise vorticity observation thus far reported, and is considered tobe particularly trustworthy because the values of P Λ ′ (cid:0) √ s NN = 200 GeV , C = 20% (cid:1) and P Λ ′ (cid:0) √ s NN = 200 GeV , C = 20% (cid:1) are considered (in [7]) to be essentially indistinguishablewith these uncertainties, and this is expected on theoretical grounds. We see that, by thesame measure, these results are also consistent with both with our vorticity bound (24) and with our conjectured equality, (25).If one repeats this calculation for collisions at an impact energy of 62.4 GeV, onefinds that the energy density is of course lower (about . / fm ), as is the temperature(about 179 MeV) and that the angular momentum density also drops, but more sharply ,14o around . / fm (it scales approximately linearly with √ s NN [38]): this pattern isseen throughout these calculations. The result is a much less stringent bound, (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 62 . , C = 20%) ≤ ≈ . , (31)which might well be detectable in an analysis similar to that of [7]; unfortunately, withthe current data the error bars are large in this case (see the second-from-rightmost pointsin Figure 4 of [6]), and clear evidence of polarization is yet to be obtained. There is inany case no conflict with our claim that P Λ ′ + P Λ ′ can be no larger than this or indeedthat (25) might be valid here.At the other extreme, one can consider the lead-lead collisions studied in the ALICEexperiment at the LHC: here, in the collisions at 2.76 TeV, the energy density [45] isabout 2.3 times larger than in the 200 GeV collisions, but the angular momentum density isabout 13.5 times larger for a given centrality; furthermore, the temperature is considerablyhigher, roughly 300 MeV. The ALICE investigation of peripheral collisions [46] consideredcentrality in two ranges: to , and also to . As before, in the first casewe can take to be representative, and then we obtain from (24) an extremely severebound: (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 2 .
76 TeV , C = 20%) ≤ ≈ . . (32)The other range is interesting, since the data go down to a very low centrality. Herethe angular momentum is enormous, but it does not vary monotonically with impactparameter, so this case merits separate investigation. The much larger overlap volumewhen the impact parameter is small (around 3.5 fm for centrality) makes itself felthere, and we find in this case a slightly less stringent bound despite the higher angularmomentum: (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 2 .
76 TeV , C = 5%) ≤ ≈ . . (33)This interesting relaxation of the bound at low centralities is characteristic of the holo-graphic model, and we will discuss it in more detail elsewhere. For the present we merelynote that this is still an extremely low value.Even with the substantial (theoretical and observational) uncertainties here, it is clearthat, in all cases, the vorticity bound is (at present) completely inconsistent with anyobservation of hyperon polarization in these experiments (and of course this prediction iseven more firm for the collisions at 5.02 TeV [47]) . This is entirely consistent with thereported data, in which no evidence of Λ / Λ polarization was detected [46].In summary, the vorticity bound asserts that global polarization of Λ and Λ hyperonsshould certainly not be observable in current data at impact energies much above 200GeV. It is consistent with a tiny total polarization at 200 GeV — now observed, atalmost exactly the maximum value permitted by the bound. If the uncertainties can bevery considerably reduced, and if (25) continues to hold, we expect it to be observable incollisions at 62.4 GeV, at a total percentage about double the observed value at 200 GeV.Let us turn, then, to much lower impact energies. That is, the vorticity is predicted to be larger for smaller angular momentum densities; this is clearfrom (24) directly, and it is in fact in agreement with all of the reported data. See [38] for the physics ofthis. Hyperon polarization may, however, be observable at very high impact energies in future, perhaps inruns 3 or 4 of the LHC [48]. .2 Collisions at 39, 27, and 19.6 GeV The STAR collaboration took data at 39, 27, and 19.6 GeV impact energies. We interruptour investigation at 19.6 GeV because, while data were also taken at still lower impactenergies (to be discussed below), it is not completely clear that the QGP is actually formedin those cases; this is discussed in detail in [39]. We will not take a stand on this issue,but we find it clearest to focus first on the cases which are not in doubt.In the case of collisions at 39 GeV, with centrality, we find that the angularmomentum density α has dropped to around . / fm , the energy density ε to . / fm ,the temperature to 178 MeV, and so the vorticity bound (24) gives us (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 39 GeV , C = 20%) ≤ ≈ . (34)the corresponding collisions at 27 GeV have α ≈ . / fm , T ≈ MeV, and ε ≈ . / fm , and so we have (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 27 GeV , C = 20%) ≤ ≈ . . (35)Finally, for collisions at 19.6 GeV we have a still lower angular momentum density ofaround . / fm , T ≈ MeV, and the energy density is about . / fm , leading to (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 19 . , C = 20%) ≤ ≈ . . (36)The agreement with Figure 4 of [6] (sixth pair from left for 39 GeV, fifth from left for27 GeV, fourth from left for 19.6 GeV) (of course one has to add the two values shownthere at each impact energy) is better than one was entitled to expect in a holographicmodel (that is, agreement to within a factor of at best 2). The rate at which the totalpolarization declines with increasing impact energy is reproduced particularly well.In short: at these impact energies, the vorticity bound relaxes quite dramatically, tothe point where global polarization of Λ and Λ hyperons should be clearly observable;and so it has proved: these are the impact energies for which the evidence for hyperonpolarization arising from QGP vorticity was most clear-cut in [6].Finally, we consider the collisions with the lowest impact energies. The reported data [6] on the Λ and Λ hyperon polarizations present a less clear picturethan in the case just considered. In particular, the Λ polarization results appear to besignificantly larger than those for Λ hyperons, and this suggests that some additionaleffect may be at work here, making the interpretation of these results somewhat dubious:see [49] and particularly [50]. In addition, at these impact energies (particularly for the7.7 GeV case), it is open to doubt whether a QGP actually forms. If this is not the case,of course, then a gauge-gravity approach cannot be used .With these warnings noted, the predictions of the holographic model are as follows.At 14.5 GeV, α ≈ . / fm , T ≈ MeV, ε ≈ . / fm , and then (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 14 . , C = 20%) ≤ ≈ . (37)16ollisions at 11.5 GeV have α ≈ . / fm , T ≈ MeV, and ε ≈ . / fm , leading to (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 11 . , C = 20%) ≤ ≈ . (38)and finally the 7.7 GeV collisions have α ≈ . / fm , T ≈ MeV, and ε ≈ . / fm ,giving (cid:2) P Λ ′ + P Λ ′ (cid:3) ( √ s NN = 7 . , C = 20%) ≤ ≈ . . (39)Except at 7.7 GeV, the agreement with [6] continues to be fairly good. In the 7.7 GeVcase, the reported polarization for Λ hyperons is so much larger than that for Λ hyperonsthat this case should be viewed with particular caution. In any event, in view of the largeerror bars in these cases, we can still assert that there is at least no contradiction to thevorticity bound.It is noteworthy that, as one proceeds to higher impact energies, the difference betweenthe reported Λ and Λ hyperon polarizations grows steadily smaller, being quite negligible[7] at 200 GeV; at the same time, the agreement of the vorticity bound, and of equation(25), with the data becomes steadily better. This may not be a coincidence.Figure 1: Theoretical upper bounds on total Λ hyperon polarization, that is, (cid:2) P Λ ′ + P Λ ′ (cid:3) (cid:0) √ s NN , C = 20% (cid:1) ≤ Φ (cid:0) √ s NN , C = 20% (cid:1) , as a percentage, for collisions at √ s NN = 7 . , . , . , . , , , . , GeV and centrality.Our results are summarized in Figure 1, which should be compared with Figure 4 of [6]and Figure 4 of [7] by adding together the values corresponding to the two points at eachimpact energy. The figures appear to be compatible.A more broad-brush way of making a comparison with the results of [6] is to computethe vorticity itself, averaged over impact energies. As mentioned above, in [6] this is givenas ± × · s − , but with a large systematic uncertainty of order 2. Here we find thatthe √ s NN -averaged value of ω , computed using (25), is approximately . × · s − ,17omewhat low, but in reasonable agreement with the data in view of the uncertainties.(The principal uncertainty is, once again, primarily associated with the difficulty [37] ofdetermining the temperatures; the temperature estimates used in [6] differ somewhat fromthose used here.)Our claim, then, is that the relation (25), inspired by the simplest possible holographicmodel of this system, approximately captures the actual relation between the vorticitiesand the angular momentum densities of the plasmas generated by peripheral collisions,at least for impact energies which are not very low (meaning below 11.5 GeV).We should also be cautious with regard to centralities. We have seen that both (24) and(25) are valid for collisions at √ s NN = 11 . GeV and centrality , with α ≈ fm − . Weshould therefore not assume that the bound is attained at any angular momentum densitybelow around fm − . This translates to an impact parameter no lower than 2.5 fm (orcentrality about 2.5 % ). On the other hand, all of our discussions have concerned collisionswhich are not very peripheral, with centrality not much greater than 20 % , correspondingto an impact parameter no greater than about 7 fm. This, then, is the domain in whichwe claim that (24) and (25) are valid.
6. Conclusion
We have studied the AdS -Kerr spacetime from a holographic point of view. Such a blackhole, with an angular momentum to mass ratio A , corresponds to matter at conformalinfinity with an angular momentum density to energy density ratio also equal to A , andwith an angular velocity which can at least be bounded above. We have conjectured thata more complete analysis, were it possible, would turn this bound into an equation, andwe have argued that the data reported by the STAR collaboration is consistent with thisconjecture; so are the corresponding results from ALICE at the LHC, in the sense thatthe non-observation of Λ / Λ hyperon polarization there is consistent with the small valuespredicted by equation (25).The applicability of holographic techniques to this problem is fundamentally limited:the no-hair theorems ensure that we have very few parameters at our disposal in the bulk.The “universality” of black hole physics is often cited [2] as a virtue of the holographicapproach, but in this case it severely restricts the number of properties of the “peripheralplasma” we can hope to represent .An optimistic assessment of these results would assert that, within its domain ofapplicability, the holographic model works unexpectedly well. The agreement of Figure 1with Figure 4 of [6] and Figure 4 of [7], apart from one possible outlier, is surprising. Thefact that the model predicts, correctly, that hyperon polarization associated with QGPvorticity should be readily observable at impact energies up to around 39 GeV, observableonly with difficulty at impact energy 200 GeV, and not at all (in current experiments)at higher energies, is very suggestive. A pessimistic assessment would assert that the The only parameter we have not used is the angular momentum corresponding to rotation of the bulkblack hole around a second axis; that is, one could use the most general metric given in [1], the metric g (cid:16) AdSK ( a,b )5 (cid:17) in our notation, where b represents a second, independent angular momentum parameter.More speculatively, one could try to use five-dimensional rotating objects with non-spherical horizontopologies, if these can be found explicitly in the asymptotically AdS context [51]. κ occurring in those relations; perhaps thiscan be improved or given a firmer basis by more sophisticated considerations. At leastwe have a concrete basis for further investigations by other methods.We have seen that holography focuses our attention on a specific parameter, theratio ε/α . This quantity depends in a complicated but definite manner on the centralityof a peripheral collision, and the dependence is particularly marked for centralities muchsmaller than those considered here (or in [6, 7]). Our considerations therefore allow pre-dictions to be made regarding what one must expect to find if data can be taken at smallcentralities. This will be discussed elsewhere. Acknowledgements
The author thanks Dr Soon Wanmei for valuable discussions.